Theories of knowledge divide naturally, theoretically and historically
into the two rival schools of rationalism and empiricism. Neither
rationalism nor empiricism disregards the primary tool of the
other school entirely. The issue revolves on beliefs about necessary
knowledge and empirical knowledge.

1. Rationalism

Rationalism believes that some ideas or concepts are independent
of experience and that some truth is known by reason alone.

a. a priori

This is necessary knowledge not given in nor dependent upon experience;
it is necessarily true by definition. For instance "black
cats are black." This is an analytic statement, and broadly,
it is a tautology; its denial would be self-contradictory.

2. Empiricism

Empiricism believes that some ideas or concepts are independent
of experience and that truth must be established by reference
to experience alone.

b. a posteriori

This is knowledge that comes after or is dependent upon experience.
for instance "Desks are brown" is a synthetic statement.
Unlike the analytic statement "Black cats are black",
the synthetic statement "Desks are brown" is not necessarily
true unless all desks are by definition brown, and to deny it
would not be self-contradictory. We would probably refer the
matter to experience.

Since knowledge depends primarily on synthetic statements --
statements that may be true or may be false -- their nature and
status are crucial to theories of knowledge. The controvercial
issue is the possibility of synthetic necessary knowledge -- that
is, the possibility of having genuine knowledge of the world without
the need to rely on experience. Consider these statements:

1) The sum of the angles of a triangle is 180 degrees.

2) Parallel lines never meet.

3) A whole is the sum of all its parts.

Rationalism may believe these to be synthetic necessary statements,
universally treu, and genunie knowledge; i.e., they are not merely
empty as the analytic or tautologous statemenst (Black cats are
black) and are not dependent on experience for their truth value.

Empiricism denies that these statements are synthetic and necessary.
Strict empriicism asserts that all such statements only appear
to be necessary or a priori. Actually, they derive from experience.

Logical empiricism admits that these statements are ncessary
but only because they are not really synthetic statements but
analytic statements, which are true by definition alone and do
not give us genuine knowledge of the world.

GENUINE KNOWLEDGE

Rationalism includes in genuine knowledge synthetic necessary
statements (or, if this term is rejected, then those analytic
necessary statements that "reveal reality" in terms
of universally necessary truth; e.g., "An entity is what
it is and not something else.")

Empiricism limits genuine knowledge to empirical statements.
Necessary statements are empty (that is, they tell us nothing
of the world).

Logical empiricism admits as genuine knowledge only analytic
necessary (Black cats are black) or synthetic empirical statements
(desks are brown). But the anyalytic necessary statements or
laws of logic and mathematics derive from arbitrary rules of usage,
definitions, and the like, and therefore reveal nothing about
reality. (This is the antimetaphysical point of view).